Discrete Subgroups Isomorphic to Lattices in Semisimple Lie Groups

Abstract

Let G be a locally compact topological group. A discrete subgroup T of G is said to be a lattice in G if the homogeneous space G/T carries a finite G invariant measure. A lattice T in G is said to be uniform if G/T is compact otherwise, it is said to be nonuniform. A lattice T in a semisimple real analytic group G is irreducible if no subgroup of T of finite index is a direct product of two infinite normal subgroups. It is well-known that given a linear analytic semisimple group G which has no compact factors and a lattice T in G, there exists a unique almost direct product decomposition G = IIGl (G{ a normal analytic subgroup of G) such that T{ = T n Gi is an irreducible lattice in Gt and (then) TTTi is a subgroup of T of finite index. Following Iwasawa, we define the characteristic index x(G) of a Lie group G, which has finitely many connected components, by